We introduce a new family of distributions to approximate $\mathbb {P}(W\in
A)$ for $A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent
integer-valued random variables $\xi_1$, $\xi_2$, $...,$$\xi_n$ with finite
second moments, where, with large probability, $W$ is not concentrated on a
lattice of span greater than 1. The well-known Berry--Esseen theorem states
that, for $Z$ a normal random variable with mean $\mathbb {E}(W)$ and variance
$\operatorname {Var}(W)$, $\mathbb {P}(Z\in A)$ provides a good approximation
to $\mathbb {P}(W\in A)$ for $A$ of the form $(-\infty,x]$. However, for more
general $A$, such as the set of all even numbers, the normal approximation
becomes unsatisfactory and it is desirable to have an appropriate discrete,
nonnormal distribution which approximates $W$ in total variation, and a
discrete version of the Berry--Esseen theorem to bound the error. In this
paper, using the concept of zero biasing for discrete random variables (cf.
Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237--260]), we introduce a
new family of discrete distributions and provide a discrete version of the
Berry--Esseen theorem showing how members of the family approximate the
distribution of a sum $W$ of integer-valued variables in total variation.Comment: Published at http://dx.doi.org/10.1214/009117906000000250 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org